Everything is Geometrical: Hermann Grassmann’s Algebra

Hermann Grassmann (1809-1877), Prussian mathematician from Stettin, in what is now Poland. Hipster gentlemen, take note of his luxuriously curly chinbeard.

Over a period of time, I’ve been gradually introducing the 18th and 19th century mathematics foundational to 20th century physics: non-Euclidean geometry, complex numbers, quaternions, and Clifford algebras. I doubt I’ll ever cover all of it, and of course I can only provide a very minimal introduction to each of these topics (without losing all my readers at least!). Today’s installment is perhaps the most important of all, though maybe even the least well-known: the algebra of Hermann Grassmann.

Grassmann was a schoolteacher, and his most important mathematical work was done during his spare time. (Knowing how much schoolteachers work today, I wonder either if his work load was lighter, or if he just neglected sleep. I suspect the latter, based on my own experience teaching college.) William Kingdon Clifford was among the first to recognize the importance of Grassmann’s work; the combination of Grassmann algebra with Hamilton’s quaternions is what we know as Clifford or geometric algebra today. However, Grassmann’s algebra as its own subject is still useful, and plays a role especially in understanding the physics of fermions: the class of particles that includes all matter as we know it!

What particle are you? Sean Carroll made the definitive flowchart for the Standard Model of particles and interactions; click for a much higher-resolution version.

Fermions are particles that obey the Pauli exclusion principle: two particles in the same physical state can’t occupy the same space. For example, that means for two electrons to have the same energy inside an atom, they must have opposite spins—no other possibility is allowed. Let’s think about how we might represent that mathematically: we’ll represent each electron as a (currently unspecified) mathematical object, and design matters to make them obey the Pauli exclusion principle. We’ll define a special kind of multiplication that changes sign if you swap the order:where ξ and η both represent electrons. If the electrons have different quantum states, we represent them with different symbols, but if they are in the same state, we use the same symbol for both, since electrons are indistinguishable from each other. The special multiplication symbol (called the “outer product”, if you want the technical term) is sometimes left out, but I like keeping it in place, to make sure we know that it’s a special kind of multiplication.

Obviously ξ and η aren’t ordinary numbers, since the order of multiplication matters. They aren’t quaternions or vectors in Clifford algebra either, since switching the order always adds a negative sign. Instead, ξ and η are known as Grassmann numbers, and when you write down the whole syntax for working with them, you get Grassmann algebra. While Grassmann didn’t know about fermions, his algebra turns out to be just what Doctor Pauli ordered. We can stack up as many electrons as we want, and interchanging any two of them changes the sign:for three electrons, andfor four electrons. (I’m not showing all the possible variations; suffice to say that the more electrons involved, the more possible swaps you can have.) But here’s the thing: if any two electrons have the same quantum state—meaning they are represented by the same Grassmann number—the whole string of multiplications is going to be zero!

We can (and usually do) make things more complicated: we can multiply Grassmann numbers by regular real and complex numbers, and add them together as well as multiply. Ordinary numbers multiply together just like you’re used to, so it doesn’t matter if we use the “∧” symbol or not:and multiplying a Grassmann number by a regular number is the same way (just as with quaternions) A general Grassmann number can look something like this:with lots of other possibilities! In fact, if you read my post about Clifford algebra, that Grassmann number should look a little familiar: Clifford built on Grassmann’s ideas to make his geometric algebra. While the Grassmann number g may not be a typical quantum state, something resembling it can show up when you start diving into the physics—which I won’t do in this post, in the interests of keeping it short.

A Little Deeper

Stroboscopic view of a thrown ball. The horizontal and vertical axes really do represent perpendicular directions.

Today, Grassmann algebra shows up in a wide variety of applications, many of which don’t even acknowledge the Chinbearded One. The reason why is that Grassmann’s math doesn’t assume anything about shape of the space you’re working in: it’s a completely general geometry. What do I mean by that? To use an example: when we plot coordinates, we draw horizontal and vertical axes (and maybe a third axis to get three dimensions in). Those axes are perpendicular to each other, which surprisingly means we’ve made an assumption: that there is a set relationship between the horizontal axis and vertical axis.

Velocity and position can be plotted on the same axes, but they aren't really perpendicular quantities. (See my earlier post for a complete explanation of this figure.)

If we’re making a map or drawing the motion of a thrown baseball, that assumption is right and fair: those quantities are perpendicular to each other in a real sense. Quaternions and Clifford numbers have a built-in perpendicularity, which means there is a metric somewhere: a measure of whether quantities are aligned or not. But what if we’re comparing two quantities that don’t describe the same physical attribute, such as the velocity and the position of a pendulum? It’s useful to draw them on a single plot, they aren’t really “perpendicular” to each other—even though they’re independent quantities. Grassmann’s approach lets you describe that kind of situation: you can assign the quantities to Grassmann numbers and use the algebra to manipulate them.

Mostly people won’t call that Grassmann algebra; they’ll give it a name like “differential forms” or “fiber bundle space” (which is one of my favorite phrases in math), but let’s give props where they are due. It’s all Grassmann, and when you think of things that way, you realize that everything is geometrical. Chemists (and physicists) draw diagrams with pressure on one axis and volume on another: they aren’t really perpendicular quantities, but they can be described with Grassmann’s math.